# Cuisenaire Rods

## CUISENAIRE RODS^{®}

#### REINFORCE KEY MATH CONCEPTS AND DEVELOP ABSTRACT THINKING

Cuisenaire Rods are a versatile collection of rectangular rods of 10 colors, each color corresponding to a different length. Provide endless opportunities to introduce, investigate, and reinforce key math topics such as addition, subtraction, geometry, measurement, multiplication, and division.

#### SKILLS

- Adding & Subtracting
- Multiplication & Division
- Fractions
- Place Value
- Counting
- Geometry & Measurement
- Ratio & Area

## Learn About Cuisenaire Rods

Cuisenaire Rods are a versatile collection of rectangular rods of 10 colors with each color corresponding to a different length. The shortest rod, the white, is 1 centimeter long; the longest, the orange, is 10 centimeters long. One set of rods contains 74 rods: 4 each of the orange (σ), blue (e), brown (n), black (k), dark green (d), and yellow (y); 6 purple (p); 10 light green (g); 12 red (r); and 22 white (w). One special aspect of the rods is that, when they are arranged in order of length in a pattern commonly called a “staircase,” each rod differs from the next by 1 centimeter, which is the length of the shortest white rod.

Unlike Color Tiles, which provide a discrete model of numbers, Cuisenaire Rods, because of their different yet related lengths, provide a continuous model. This means they allow you to assign a value to 1 rod and then assign values to the other rods by using the relationships among the rods.

Cuisenaire Rods can be used to develop a wide variety of mathematical ideas at many different levels of complexity. Initially, however, students use the rods to explore spatial relationships by making flat designs that lie on a table or by making three-dimensional designs by stacking the rods. The intent of students’ designs, whether to cover a certain amount of a tabletop or to fill a box, will lead students to discover how some combinations of rods are equal in length to other, single rods. Students’ designs can also provide a context for investigating symmetry. Older students who have no previous experience with Cuisenaire Rods may explore by comparing and ordering the lengths of the rods and then recording the results on grid paper to visualize the inherent “structure” of the design. In all their early work with the rods, students have a context in which to develop their communication skills through the use of grade-appropriate arithmetic and geometric vocabulary.

Though students need to explore freely, some may appreciate specific challenges, such as being asked to make designs with certain types of symmetry or certain characteristics, such as different colors representing different fractional parts.

One of the basic uses of Cuisenaire Rods is to provide a model for the numbers 1 to 10. If the white rod is assigned the value of 1, the red rod is assigned the value of 2 because the red rod has the same length as a “train” of two white rods. Similarly, the rods from light green through orange are assigned values from 3 through 10, respectively. The orange and white rods can provide a model for place value. To find the length of a certain train, students can cover the train with as many orange rods as they can and then fill in the remaining distance with white rods; so a train covered with 3 orange rods and 4 white rods is 34 white rods long. The rods can be placed end to end to model addition. For example 2 plus 3 can be found by first making a train with a red rod (2) and a light green rod (3) and then finding the single rod (yellow) whose length (5) is equal in length to the 2-rod train. This model corresponds to addition on a number line.

The rods can also be used for acting out subtraction as the search for a missing addend. For example, 5 minus 2 can be found by placing a red rod (2) on top of a yellow (5), then looking for the rod which, when placed next to the red, makes a train equal in length to the yellow.

Multiplication, such as 5 times 2, is interpreted as repeated addition by making a train of 5 red rods or of 2 yellow rods.

Division, such as 10 divided by 2, may be interpreted as repeated subtraction (“How many red rods make a train as long as an orange rod?”) or as sharing (“Two of what color rod make a train as long as an orange rod?”).

Cuisenaire Rods also make effective models for decimals and fractions. If the orange rod is designated as the unit rod, then the white, red, and light green rods represent 0.1, 0.2, and 0.3, respectively. If the dark green rod is chosen as the unit, then the white, red, and light green rods represent 1⁄6, 2⁄6 (1⁄3), and 3⁄6 (1⁄2), respectively. Once the unit rod has been established, addition, subtraction, multiplication, and division of decimals and fractions can be modeled in the same way as the operations with whole numbers.

Cuisenaire Rods are suitable for a variety of geometric and measurement investigations. Once students develop a sense that the white rod is 1 centimeter long, they have little difficulty in accepting and using centimeters as units of length. Since the face of the white rod has an area of 1 square centimeter, the rods are ideal for finding area in square centimeters. Since the volume of the white rod is 1 cubic centimeter, the rods can exemplify the meaning of volume as students use rods to fill up boxes. Students may even develop a sense of a milliliter as the capacity of a container that holds exactly one white rod. Cuisenaire Rods offer many possibilities for forming and discovering number patterns both through creating designs that are growing according to some pattern and through finding the number of ways in which a rod can be made as the sum of other rods. This second scenario can lead to the concept of factors of a number and prime numbers.

The rods also provide a context for building logical reasoning skills. For example, students can use 2 loops of string to create a Venn Diagram showing the multiples of both red and light green rods (which represent 2 and 3, respectively) by placing the multiples of red (red, purple, dark green, brown, and orange) in 1 loop, the multiples of light green (light green, dark green, and blue) in the other loop

Cuisenaire Rods are wonderful tools for assessing students’ mathematical thinking. Watching students work with Cuisenaire Rods gives you a sense of how they approach a mathematical problem. Their thinking can be “seen”, in that thinking is expressed through the way they construct, recognize, and continue spatial patterns. When a class breaks up into small working groups, you are able to circulate, listen, and raise questions, all the while focusing on how individuals are thinking. Here is a perfect opportunity for authentic assessment. Having students describe their designs and share their strategies and thinking with the whole class gives you another opportunity for observational assessment. Furthermore, you may want to gather students’ recorded work or invite them to choose pieces to add to their math portfolios.

## How to use Cuisenaire Rods

Cuisenaire Rods are extremely versatile. Use them to teach concepts from counting to fractions. See the MANY ways to use Cuisenaire Rods in grades K-5!

## Lessons

Jumbo Cuisenaire Rods are easier to build with for very young students. As they explore they naturally learn about relationships between the rods. Allowing time for all students to gain familiarity with Cuisenaire Rods is essential at this stage. You can do this by providing daily play time with the rods for up to a week or even longer. Play time is especially important after the students have begun to use the rods to learn math, so that the natural engagement of hands-on manipulatives continues.

Rods can be used to build understanding of addends. Students look for various ways to make trains equal in length to a specific rod (for example, the light green rod). This activity lets them see how a number can be represented in different ways.

Here are 2 activities to help students get to know Cuisenaire Rods and to begin using them to learn and understand the math that they represent.

Activity | Curriculum Strands | Topics |
---|---|---|

Alike and Different
Students make shapes from Cuisenaire Rods of their own choosing. Then they compare their shapes and discuss how the shapes are alike and how they are different. |
Number |
Comparing, Counting, Spatial Visualization |

Challenge Match
In this game for two players, students take turns matching 2-rod Cuisenaire Rods trains to a single rod in an effort to be the last to make a 2-rod train. |
Number Logic | Game Strategies |

In intermediate grades, students can explore Cuisenaire Rods as they model multiplication and division. Rods can also be used to model fractions by treating any single rod as the unit length and finding the fractional names of other rods. You should encourage students to reason about the value of each rod based on its relationship to other rods and continue to provide time for free exploration on a regular basis.

Here are two activities using Cuisenaire Rods in different contexts.

Activity | Curriculum Strands | Topics |
---|---|---|

Shopping for Rods
Students use a spinner to find the cost of 1 Cuisenaire Rod. They use this value to figure out the cost of other rods and then determine a combination of rods that they could buy with $10.00. |
Number |
Counting, Ratio & Proportion, Dealing with Money |

Naming Rods
Students assign a value of 1 whole unit to a Cuisenaire Rod of their choice. They then identify each of the other rods as a number based on its relationship to the unit rod. |
Number |
Fractions, Comparing, Looking for Patterns |

Rob Stamping
Children will explore surface area and use spatial reasoning to make predictions about the surface area of any size rod. ## View Teacher Guide |
Measurement, Patterns/Functions |
Spatial Visualization, Surface Area, Growth Patterns |

In middle grades, Cuisenaire Rods are an excellent tool for modeling ratios and proportional relationships. Students can also use the rods to create patterns and then describe those patterns using a variety of mathematical representations, including charts, graphs, and equations.

Here are 2 activities using Cuisenaire Rods in different contexts.

Activity | Curriculum Strands | Topics |
---|---|---|

Blueprints
Students build structures with Cuisenaire Rods and draw the different two-dimensional views of each—front, back, left, right, top, and bottom. Then they reverse the process by using one another’s drawings to build matching structures. |
Number |
Counting, Ratio & Proportion, Dealing with Money |

Building Pyramids
Students use Cuisenaire Rods to build models of pyramids. They record data about each structure, look for patterns, and make conjectures. |
Number |
Fractions, Comparing, Looking for Patterns |

Growing Everyday
Students will create and extend growth patterns and make and check predictions based on those patterns. ## View Teacher Guide |
Number, Patterns/Functions |
Spatial Visualization, Looking for Patterns, Comparing |